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| Crank–Nicolson | |
|---|---|
| Name | Crank–Nicolson method |
| Developer | John Crank, Phyllis Nicolson |
| Introduced | 1947 |
| Field | Numerical analysis |
| Related | Finite difference method, Heat equation, Partial differential equation |
Crank–Nicolson The Crank–Nicolson method is an implicit time-stepping scheme for solving parabolic Partial differential equations, introduced by John Crank and Phyllis Nicolson in 1947. It combines ideas from the Galerkin method and the Finite difference method to yield a time-centered, second-order accurate integrator frequently used in Computational fluid dynamics, Financial mathematics, and Heat conduction. The scheme is notable for its balance of accuracy and stability, and for connections to methods developed by Lewis Fry Richardson, Richard Courant, and Kurt Friedrichs.
The method applies to initial-boundary value problems such as the Heat equation, where temporal discretization parallels spatial discretizations used in Thomas algorithm-amenable tridiagonal systems. Its formulation builds on historical work by John von Neumann on stability analysis, and it sits alongside contemporaneous techniques from researchers like Hermann Weyl and Andrey Kolmogorov. Practitioners in Numerical linear algebra and teams at institutions such as Argonne National Laboratory, Los Alamos National Laboratory, and Lawrence Livermore National Laboratory have employed the scheme in large-scale simulations. The Crank–Nicolson approach influences software libraries including LAPACK, PETSc, Trilinos, and SciPy.
Starting from a prototype parabolic PDE (the Heat equation) one discretizes time using the midpoint rule, which is equivalent to the trapezoidal rule in time. The spatial second derivative is approximated via centered finite differences akin to schemes described by Carl Friedrich Gauss and Adrien-Marie Legendre in earlier numerical work, producing a linear system at each time level. Applying the method to a one-dimensional diffusion problem yields a tridiagonal matrix closely related to matrices studied by Thomas (algorithm), with coefficients that mirror stability analyses by John von Neumann and consistency arguments as in the work of Richard Courant. The derivation parallels variational interpretations used by Ivo Babuška and connections to the Crank–Nicolson developers’ original report linking to techniques of Rayleigh and Ritz.
The scheme is second-order accurate in time and, with centered spatial differences, second-order accurate in space, comparable to methods championed by Gustav Kirchhoff and Pierre-Simon Laplace for diffusion analogues. It is A-stable in the linear setting, a property analyzed in the tradition of stability theory advanced by Earl Coddington and Norman Levinson, and closely related to the implicit trapezoidal rule studied by J. L. Lions and Claude Brezinski. For linear constant-coefficient problems the amplification factors tie to analyses by John von Neumann and Oskar Perron. In practice, oscillatory initial data evoke considerations similar to those addressed by Andrey Markov (mathematician) and Marcel Riesz.
Implementation typically assembles a banded linear system solved each timestep via the Thomas algorithm for tridiagonal matrices or via iterative solvers from Krylov subspace methods such as Conjugate gradient method for symmetric problems and GMRES for nonsymmetric ones. Preconditioning strategies draw on research at Argonne National Laboratory and methods like Incomplete LU factorization from Youcef Saad’s body of work. Time-stepping control can leverage adaptive stepping informed by error estimators used in Runge–Kutta methods and software patterns from PETSc and Trilinos. Parallel implementations exploit domain decomposition methods associated with Alan Turing’s early computation ideas and later frameworks from MPI-based projects at National Energy Research Scientific Computing Center.
Widely used in modeling Heat conduction and transient diffusion in engineering problems studied at Massachusetts Institute of Technology, Stanford University, and Imperial College London, it is also standard in pricing Black–Scholes derivatives in Financial mathematics researched at Princeton University, University of Chicago, and Goldman Sachs quant groups. In Computational fluid dynamics the scheme appears in simplified advection–diffusion contexts considered by researchers at NASA and European Space Agency. Other applications include neutron diffusion in reactors analyzed at Oak Ridge National Laboratory, groundwater transport studies from United States Geological Survey, and option pricing models developed in finance groups at Barclays and Morgan Stanley.
Extensions include operator-splitting methods related to Peaceman–Rachford and Douglas–Rachford schemes, links to higher-order backward differentiation formulas studied by Charles Bell and multistep techniques from G. Dahlquist, and complex-symmetric generalizations used in Quantum mechanics simulations influenced by Paul Dirac. Variants introduce flux limiters inspired by Boris and Book for hyperbolic corrections, weighted Crank–Nicolson families connected to θ-methods examined by Todd R. Bewley, and stabilized forms employing artificial diffusion as in the work of Jameson and Hirsch (computational fluid dynamics). Hybridizations with Finite element method lead to implicit-explicit (IMEX) integrators studied at INRIA and CWI.
Stability proofs invoke Fourier (von Neumann) analysis originally developed by John von Neumann and extended by Hille and Phillips; energy estimates mirror techniques from Lax–Milgram frameworks and the Lax equivalence theorem proven by Peter Lax and Ralph Phillips. Local truncation error is O(Δt^2) in time and O(Δx^2) in space for smooth solutions, while global error bounds derive from Grönwall-type inequalities reminiscent of work by Thomas Grönwall. For stiff or nonlinear problems, modified equation analysis and backward error analysis from Nicholas Higham and Johannes Stoer guide timestep selection and iterative solver tolerances. In practical settings, researchers at National Institute of Standards and Technology and SIAM conferences report on robustness and pathological oscillations mitigated by filtering or adaptive damping schemes championed by LeVeque and Anderson (numerical methods).